height, time, speed

[red and white kop!]

this has to do with diffrentiation and not kinematics
a boy stands on the edge of a cliff of height 60m. he throws a stone vertically upwards so that its distance h metres above the cliff top is given by h=20t -5(t^2)
a. calculate the max height of the stone above the cliff top
i differentiated h=f(t) and got 20m
b. calculate the time which elapses before the stone hits the beach (it just misses the boy on the way down (lucky him))
i took h=-60 and got t=6
c. calculate the speed with which the stone hits the beach.
here i'm lost. shouldnt speed=f'(t)=20-10t? but i know this is wrong cos the answer is 40ms^-1

 

[soroban]

Hello, red and white kop!

A boy stands on the edge of a cliff of height 60m.
He throws a stone vertically upwards so that its distance h metres above the cliff top is given by: .\(\displaystyle h\:=\:20t -5t^2\)

a. Calculate the max height of the stone above the cliff top.
. . . i differentiated \(\displaystyle h\,=\,f(t)\) and got 20m . . . . Yes!

b. Calculate the time which elapses before the stone hits the beach.
. . . i took \(\displaystyle h=-60\) and got \(\displaystyle t=6\) . . . . Right!

c. Calculate the speed with which the stone hits the beach.
. . . Here i'm lost.
. . . Shouldn't speed \(\displaystyle =\,f'(t)\,=\,20-10t\) ?
. . . . . Yes, this is the speed at any time t.

. . . but i know this is wrong, cos the answer is 40 m/s
. . . . . So what is the speed when t = 6 ?

 

[red and white kop!]

!

 

[BigGlenntheHeavy]

\(\displaystyle s(t) \ = \ -5t^2+20t+60, \ s(t) \ = \ distance \ in \ meters.\)

\(\displaystyle v(t) \ = \ -10t+20, \ v \ = \ velocity \ in \ meters/sec.\)

\(\displaystyle a(t) \ = \ -10. \ a \ = \ acceleration \ in \ meters/sec^2, \ (Not \ needed \ here).\)

\(\displaystyle a) \ v(t) \ = \ 0 \ = \ -10t+20, \ \implies \ t \ = \ 2 \ sec.\)

\(\displaystyle s(2) \ = \ -5(4)+20(2)+60 \ = \ 80m \ above \ beach, \ 20m \ above \ cliff.\)

\(\displaystyle b) \ s(t) \ = \ 0 \ = \ 5t^2-20t-60 \ = \ t^2-4t-12 \ = \ (t-6)(t+2), \ t \ = \ 6 \ sec..\)

\(\displaystyle c) \ v(6) \ = \ -60+20 \ = \ -40m/sec. \ (hits \ the \ beach), \ speed \ = \ 40m/sec.\)